Results 81 to 90 of about 648,405 (147)

Estimates for smooth Weyl sums on minor arcs

Bulletin of the London Mathematical Society, Volume 57, Issue 3, Page 657-668, March 2025.
Abstract We provide new estimates for smooth Weyl sums on minor arcs and explore their consequences for the distribution of the fractional parts of αnk$\alpha n^k$. In particular, when k⩾6$k\geqslant 6$ and ρ(k)$\rho (k)$ is defined via the relation ρ(k)−1=k(logk+8.02113)$\rho (k)^{-1}=k(\log k+8.02113)$, then for all large numbers N$N$ there is an ...
Jörg Brüdern, Trevor D. Wooley
wiley   +1 more source

Ground states of a non‐local variational problem and Thomas–Fermi limit for the Choquard equation

Journal of the London Mathematical Society, Volume 111, Issue 3, March 2025.
Abstract We study non‐negative optimisers of a Gagliardo–Nirenberg‐type inequality ∫∫RN×RN|u(x)|p|u(y)|p|x−y|N−αdxdy⩽C∫RN|u|2dxpθ∫RN|u|qdx2p(1−θ)/q,$$\begin{align*} & \iint\nolimits _{\mathbb {R}^N \times \mathbb {R}^N} \frac{|u(x)|^p\,|u(y)|^p}{|x - y|^{N-\alpha }} dx\, dy\\ &\quad \leqslant C{\left(\int _{{\mathbb {R}}^N}|u|^2 dx\right)}^{p\theta } {\
Damiano Greco, Yanghong Huang, Zeng Liu, Vitaly Moroz   +3 more
wiley   +1 more source

Affine Hardy--Littlewood--Sobolev inequalities

, 2023
Sharp affine Hardy--Littlewood--Sobolev inequalities for functions on $\mathbb R^n$ are established, which are significantly stronger than (and directly imply) the sharp Hardy--Littlewood--Sobolev inequalities by Lieb and by Beckner, Dou, and Zhu.
Haddad, Julián, Ludwig, Monika
core  

Existence, symmetry, and regularity of ground states of a nonlinear Choquard equation in the hyperbolic space

Advances in Nonlinear Analysis
In this study, we explore the positive solutions of a nonlinear Choquard equation involving the Green kernel of the fractional operator (−ΔBN)−α⁄2{\left(-{\Delta }_{{{\mathbb{B}}}^{N}})}^{-\alpha /2} in the hyperbolic space, where ΔBN{\Delta }_{{{\mathbb{
Gupta Diksha, Sreenadh Konijeti
doaj   +1 more source

Nonexistence and existence of solutions for a Choquard–Kirchhoff type equation involving mixed local and nonlocal operators

Electronic Journal of Qualitative Theory of Differential Equations
In this paper, we study the following Choquard-Kirchhoff type equation $$ \left( a+b\|u\|^{p(\theta-1)}\right) \left( -\Delta_{p}u+(-\Delta)^{s}_{p}u\right) =\left( \int_{\mathbb R^N}\frac{|u(y)|^{p_{\mu}^{*}}}{|x-y|^\mu}dy\right) |u|^{p_{\mu}^{*}-2}u+\
Wenjing Chen, Jingran Feng
doaj   +1 more source

Global second‐order estimates in anisotropic elliptic problems

Proceedings of the London Mathematical Society, Volume 130, Issue 3, March 2025.
Abstract This work deals with boundary value problems for second‐order nonlinear elliptic equations in divergence form, which emerge as Euler–Lagrange equations of integral functionals of the Calculus of Variations built upon possibly anisotropic norms of the gradient of trial functions.
Carlo Alberto Antonini, Andrea Cianchi, Giulio Ciraolo, Alberto Farina, Vladimir Maz'ya   +4 more
wiley   +1 more source

On the integral systems related to Hardy-Littlewood-Sobolev inequality [PDF]

Mathematical Research Letters, 2007
We prove all the maximizers of the sharp Hardy-Littlewood-Sobolev inequality are smooth. More generally, we show all the nonnegative critical functions are smooth, radial with respect to some points and strictly decreasing in the radial direction.
openaire   +2 more sources

A Framework for Proving Hilbert's Double Integral Inequality and Related Results [PDF]

, 1999
The classical Hilbert and Hardy integral inequalities are derived within a functional analytic framework.
Peachey, Tom C
core   +4 more sources

Ground state solution for Schrödinger-Choquard equation: doubly critical case

Boundary Value Problems
In this paper, we investigate the following Schrödinger-Choquard equation: − Δ u + u = ( I α ∗ | u | 2 α ♯ ) | u | 2 α ♯ − 2 u + | u | q − 2 u + | u | r − 2 u , x ∈ R N , $$ -\Delta u+u = (I_{\alpha }*|u|^{2_{\alpha }^{\sharp }})|u|^{2_{\alpha }^{ \sharp
Yusheng Shen, Zhiwei Zou, You Gao
doaj   +1 more source

Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains

Advances in Nonlinear Analysis, 2019
The paper is concerned with the existence and multiplicity of positive solutions of the nonhomogeneous Choquard equation over an annular type bounded domain.
Goel Divya, Sreenadh Konijeti
doaj   +1 more source